Almost global existence for the Prandtl boundary layer equations

TL;DR

This paper proves that solutions to the Prandtl boundary layer equations exist for a very long time when initial data is close to a stable profile, under specific regularity conditions.

Contribution

It establishes almost global existence of solutions to the Prandtl equations for initial data near stable profiles with real-analytic and weighted Sobolev regularity.

Findings

Solutions exist up to at least exponential time in terms of initial proximity.

The result applies to initial data close to stable profiles in a weighted Sobolev space.

The proof extends the lifespan of solutions significantly beyond local existence.

Abstract

We consider the Prandtl boundary layer equations on the half plane, with initial datum that lies in a weighted $H^1$ space with respect to the normal variable, and is real-analytic with respect to the tangential variable. The boundary trace of the horizontal Euler flow is taken to be a constant. We prove that if the Prandtl datum lies within $\varepsilon$ of a stable profile, then the unique solution of the Cauchy problem can be extended at least up to time $T_\varepsilon \geq \exp(\varepsilon^{-1}/ \log(\varepsilon^{-1}))$.